Second-order AAA algorithms for structured data-driven modeling

TL;DR

This paper introduces three new second-order structured data-driven modeling methods for dynamical systems, leveraging frequency domain data and extending the Adaptive Antoulas-Anderson algorithm to better incorporate physical differential structures.

Contribution

The paper develops and analyzes three second-order structured modeling approaches based on the barycentric form, extending existing algorithms to improve physical interpretability and accuracy.

Findings

Structured methods outperform unstructured models in numerical examples.

Proposed algorithms balance computational speed and modeling accuracy.

Theoretical analysis supports expected performance improvements.

Abstract

The data-driven modeling of dynamical systems has become an essential tool for the construction of accurate computational models from real-world data. In this process, the inherent differential structures underlying the considered physical phenomena are often neglected making the reinterpretation of the learned models in a physically meaningful sense very challenging. In this work, we present three data-driven modeling approaches for the construction of dynamical systems with second-order differential structure directly from frequency domain data. Based on the second-order structured barycentric form, we extend the well-known Adaptive Antoulas-Anderson algorithm to the case of second-order systems. Depending on the available computational resources, we propose variations of the proposed method that prioritize either higher computation speed or greater modeling accuracy, and we present a theoretical analysis for the expected accuracy and performance of the proposed methods. Three numerical examples demonstrate the effectiveness of our new structured approaches in comparison to classical unstructured data-driven modeling.